Re: An exercise about limsup/liminf of a subset sequence

You seem a little confused about what an infimum or supremum is. It is a greatest lower bound or least upper bound respectively. You had an example before where you were looking at a set of sets. A set of sets can be viewed as a poset where the partial order is containment. if . In that context, an infimum or supremum will be a set.

In the context of the current problem, you do not have a set of sets. You have a set of points of the real line. In this context, a greatest lower bound or a least upper bound will be a point, not a set. So, the first answer is not the set . How do we know that is the infimum of the set? We know it is a lower bound since for all . Given any , which shows that is not a lower bound. Hence, 0 is the greatest lower bound, and therefore the infimum for each . So, is an infinite sequence of zeros. The limit of an infinite sequence of zeros is zero. That is what is meant by the limit inferior ( ). So, when I say 0 for E', I mean 0, not {0}.

Re: An exercise about limsup/liminf of a subset sequence

Using that definition, you are probably correct about .

Here is one possible approach to a formal proof:

Obviously, , so those points cannot be in or .

If , then for all . Given , suppose that for every , there exists with . Then, obviously for any . That obviously implies . So, if you can show that given any and any , there exists with , then you have proven .

For , if for all , there exists with , then . (This condition is dual to in a similar way to how intersections are dual to unions).